A few months ago I asked a question about asking very specific questions about a paper. Now, I would really like to ask for a general overview and explanation of the main ideas of an entire paper.

Since I think it is relevant, let me mention the paper in question. I am trying to read Lawvere's Categories of space and quantity. This paper does not contain a single formula or mathematical symbol (except maybe two or three numbers). It is (in my opinion) extremely terse both in language and mathematical content. On the other hand, the bits of it I can understand are fascinating.

Now, I really don't think posting dozens of questions about specific sentences from the paper makes much sense, especially since many of them are in a sense philosophical, and at the very least answers would be opinion based. I do however feel that an expert could (if inclined) explain what Lawvere is trying to say in a couple of paragraphs - especially the philosophical gist of things.

Is it okay to ask something like "what is the essence of this article by Lawvere?" on MO?

Answers like "you're not ready to read it yet" and "it'll be obvious when you understand" are (I think) not very helpful.

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    $\begingroup$ I think a question asking for the explanation of an entire paper is rather not suitable for the format of this site. $\endgroup$
    – Stefan Kohl Mod
    May 29, 2016 at 22:49
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    $\begingroup$ I'm not sure I've ever set eyes on that paper (I might have, but reading Lawvere generally takes persistence and effort and I don't think I've spent any real time on it). Your meta question is a little tricky from a precedent-setting perspective: I can tell that you are serious, but there is potential for abuse if it were coming from a different type of learner. I almost want to suggest that you try your luck at the nForum instead. There's a good chance that someone like Urs Schreiber has spent some time on it. $\endgroup$
    – Todd Trimble Mod
    May 29, 2016 at 22:50
  • $\begingroup$ @ToddTrimble I've never used the nForum before, and I'm low level, so I feel my inputs would be almost spam there. Would it be reasonable to send Urs Schreiber a polite email? $\endgroup$
    – Arrow
    May 30, 2016 at 8:26
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    $\begingroup$ Did you look at the page ncatlab.org/nlab/show/space+and+quantity, or the paper emis.de/journals/TAC/reprints/articles/8/tr8.pdf? The former describes the latter as "a kind of review" of "Categories of space and quantity" $\endgroup$ May 30, 2016 at 8:41
  • $\begingroup$ @NeilStrickland I tried reader the latter a while ago without much success. I'll try agani. Thanks. $\endgroup$
    – Arrow
    May 30, 2016 at 9:12
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    $\begingroup$ I assure you that your honest questions at the nForum would be welcomed, especially any that you can formulate in response to reading the nLab page that Neil mentioned. The nLab is (in principle) meant to be readable for a very general audience, and feedback on what people find hard to read is useful (and very far from spam). You could try writing Urs privately as well; if you do, consider giving him permission to quote from your email at the nForum. Finally, we're all human beings here, trying to learn from each other. $\endgroup$
    – Todd Trimble Mod
    May 30, 2016 at 14:40
  • $\begingroup$ "what is the essence of this article by Lawvere?" I don't know, but shouldn't this type of request usually be answered by looking at reviews, e.g. on Mathematical Reviews (MathSciNet) or zbMATH (Zentralblatt)? $\endgroup$
    – Earthliŋ
    Jun 7, 2016 at 20:24

1 Answer 1


If it were possible to summarize a paper so briefly, chances are someone has already done it.

I see your question in part as asking for a royal road to understanding, and in part as asking for a Readers Digest version of the paper. In truth, Lawvere wrote the paper for other purposes, and probably has a lot of ideas to communicate which are not summarized without major mental effort. If they are, some editorial choices have to be made, and usually people do not volunteer to make such choices for you.

A reasonable variation on your question (and one which you should ask of your advisor or whoever is guiding your studies and recommending you read the paper) is what are some of the major takeaways other readers got out of the paper. For different readers the answer is likely to be different, and in asking the question you have to be prepared to deal with these other filters. However, even such summary perspectives do not excuse you from building your own.

I think the variation is a reasonable question to ask, but is not a good question for MathOverflow. A good question for MathOverflow would be as follows: "I have started reading paper X about Z in order to help my study of Y (specifically, I'm thinking about specific problem P). Are there aspects of the paper that might help me with problem P? Even if no application to P is visible, what are some of the takeaways of X and related questions Q that X might address (the closer to P the better)?"

Although answers to this version might be subjective, I see them as a good kind of subjective: anyone else who wants to approach X may in the answers see enough of a review of X to help decide if X is an appropriate paper for them. Or people who are interested in P (or maybe even Q) may be interested in knowing about X. As your version of the question stands, the answers may be too general and opinion-based to be useful to later readers.

Gerhard "Objectively Opines Upon Subjective Suggestions" Paseman, 2016.05.29.

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    $\begingroup$ There's nobody guiding my studies, and I don't really have anyone I can ask category theory (not to mention philosophical aspects of it). I am really not reading this paper in relation with any problem and do just want to get an understanding of Lawvere's philosophy. I guess I'm at an impasse for now. $\endgroup$
    – Arrow
    May 30, 2016 at 8:25
  • $\begingroup$ You don't have to be at an impasse. There are other people and other ways to ask. MathOverflow is not the forum for discussion though. You may find other Internet resources to help. Use MathOverflow for specific pointed questions. Gerhard "Study Groups Are Out There" Paseman, 2016.05.30. $\endgroup$ May 30, 2016 at 15:36
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    $\begingroup$ For what purposes did Lawvere write the paper? $\endgroup$
    – Arrow
    May 31, 2016 at 7:53
  • $\begingroup$ I haven't read the paper, so I don't know. Having seen glimpses of some of his work, my guess is he wanted to record a collection of ideas for the use of himself and others. At a certain level, it is important to get the technical details down and leave the pedagogy (and peer review of the ideas) for future generations. I could say "You're not ready to read it.", but that might be wrong. I think it is safe to say there are many who are not ready to read it. Gerhard "Isn't Ready To Read It" Paseman, 2016.05.31. $\endgroup$ May 31, 2016 at 19:15
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    $\begingroup$ To work ones way into Lawvere's ideas, it may help to start with his textbooks Conceptual Mathematics and Sets for Mathematics. Diving straight into his papers is bound to be difficult. But they are extremely rewarding. At least the ones I've understood so far. :-) $\endgroup$
    – John Baez
    May 31, 2016 at 21:50

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